Noncentral beta distribution

Noncentral Beta

Notation	Beta(α, β, λ)
Parameters	α > 0 shape (real) β > 0 shape (real) λ >= 0 noncentrality (real)
Support	$x \in [0; 1]\!$
PDF	(type I) $\sum_{j = 0}^{\infty} e^{-\lambda/2} \frac{\left(\frac{\lambda}{2}\right)^j}{j!}\frac{x^{\alpha + j - 1}\left(1-x\right)^{\beta - 1}}{\mathrm{B}\left(\alpha + j,\beta\right)}$
CDF	(type I) $\sum_{j = 0}^{\infty} e^{-\lambda/2} \frac{\left(\frac{\lambda}{2}\right)^j}{j!} I_x \left(\alpha + j,\beta\right)$
Mean	(type I) $e^{-\frac{\lambda}{2}}\frac{\Gamma\left(\alpha + 1\right)}{\Gamma\left(\alpha\right)} \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha + \beta + 1\right)} {}_2F_2\left(\alpha+\beta,\alpha+1;\alpha,\alpha+\beta+1;\frac{\lambda}{2}\right)$ (see Confluent hypergeometric function)
Variance	(type I) $e^{-\frac{\lambda}{2}}\frac{\Gamma\left(\alpha + 2\right)}{\Gamma\left(\alpha\right)} \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha + \beta + 2\right)} {}_2F_2\left(\alpha+\beta,\alpha+2;\alpha,\alpha+\beta+2;\frac{\lambda}{2}\right) - \mu^2$ where $\mu$ is the mean. (see Confluent hypergeometric function)

In probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a generalization of the (central) beta distribution.

The noncentral beta distribution (Type I) is the distribution of the ratio

$X = \frac{\chi^2_m(\lambda)}{\chi^2_m(\lambda) + \chi^2_n},$

where $\chi^2_m(\lambda)$ is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter $\lambda$, and $\chi^2_n$ is a central chi-squared random variable with degrees of freedom n, independent of $\chi^2_m(\lambda)$. In this case, $X \sim \mbox{Beta}\left(\frac{m}{2},\frac{n}{2},\lambda\right)$

A Type II noncentral beta distribution is the distribution of the ratio

$Y = \frac{\chi^2_n}{\chi^2_n + \chi^2_m(\lambda)},$

where the noncentral chi-squared variable is in the denominator only. If $Y$ follows the type II distribution, then $X = 1 - Y$ follows a type I distribution.

Cumulative distribution function

The Type I cumulative distribution function is usually represented as a Poisson mixture of central beta random variables:

$F(x) = \sum_{j=0}^\infty P(j) I_x(\alpha+j,\beta),$

where λ is the noncentrality parameter, P(.) is the Poisson(λ/2) probability mass function, \alpha=m/2 and \beta=n/2 are shape parameters, and $I_x(a,b)$ is the incomplete beta function. That is,

$F(x) = \sum_{j=0}^\infty \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}I_x(\alpha+j,\beta).$

The Type II cumulative distribution function in mixture form is

$F(x) = \sum_{j=0}^\infty P(j) I_x(\alpha,\beta+j).$

Algorithms for evaluating the noncentral beta distribution functions are given by Posten and Chattamvelli.

Probability density function

The (Type I) probability density function for the noncentral beta distribution is:

$f(x) = \sum_{j=0}^\infin \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}\frac{x^{\alpha+j-1}(1-x)^{\beta-1}}{B(\alpha+j,\beta)}.$

where $B$ is the beta function, $\alpha$ and $\beta$ are the shape parameters, and $\lambda$ is the noncentrality parameter. The density of Y is the same as that of 1-X with the degrees of freedom reversed.

Related distributions

Transformations

If $X\sim\mbox{Beta}\left(\alpha,\beta,\lambda\right)$, then $\frac{\beta X}{\alpha (1-X)}$ follows a noncentral F-distribution with $2\alpha, 2\beta$ degrees of freedom, and non-centrality parameter $\lambda$.

If $X$ follows a noncentral F-distribution $F_{\mu_{1}, \mu_{2}}\left( \lambda \right)$ with $\mu_{1}$ numerator degrees of freedom and $\mu_{2}$ denominator degrees of freedom, then $Z = \cfrac{\cfrac{\mu_{2}}{\mu_{1}}}{\cfrac{\mu_{2}}{\mu_{1}} + X^{-1} }$ follows a noncentral Beta distribution so $Z \sim \mbox{Beta}\left(\frac{1}{2}\mu_{1},\frac{1}{2}\mu_{2},\lambda\right)$. This is derived from making a straight-forward transformation.

Special cases

When $\lambda = 0$, the noncentral beta distribution is equivalent to the (central) beta distribution.

References

• M. Abramowitz and I. Stegun, editors (1965) "Handbook of Mathematical Functions", Dover: New York, NY.
• Christian Walck, "Hand-book on Statistical Distributions for experimentalists."